Moschini, GianCarlo, "Nash equilibrium in competitive games: live play in soccer", Economics Letters, 85: 365-371 (2005).

This paper uses game theory to model a common soccer scenario: whether to beat a goalkeeper by shooting at the near post or the far post.  The author examines goals scored in a Serie A season and determines that this model is a good descriptor of such goal-scoring scenarios and vindicates some of the conventional wisdom on the post that the goalkeeper should defend and that which the striker should attack.

Now if you're intrigued, please read my review of the paper after the jump.  Comments are welcome.

GianCarlo Moschini is a professor of economics at Iowa State University and, at the time this journal article was published, in training for his US Youth Soccer National coaching license.  One item of conventional wisdom in soccer training — that in one-on-one situations in the run of play where the striker is approaching goal at an angle, a goalkeeper should defend the near post and a striker should attack the far post — serves as motivation for studying the effectiveness of a certain mathematical model.  This particular model falls under the branch of game theory.

Game theory is an area of applied mathematics that attempts to study strategic interactions between objects, the decisions or strategies made, and their resulting payoffs.  This field has been widely studied in economics (eight Nobel Prizes in Economics have been awarded for research in game theory), but has also found application to problems in political science, mathematical control theory, computer science, and biology.  There are different types of game classifications, such as those where there is only one "winner" (a zero-sum game), the strategy taken is independent of the player (a symmetric game), or the players are allowed to make binding agreements (a cooperative game). 

It's common to want to find the optimum strategy in a game — the one that will either maximize the payoff to one participant or the other, or cause no difference to the payoff if a slight variation to the strategy is made.  The latter is called an equilibrium, and the most famous kind of equilibrium is a Nash equilibrium, named after economist John Nash (yes, the same one from the movie A Beautiful Mind).  A Nash equilibrium is a situation where all the players in a game know the equilibrium strategies of the other players and no one gains by unilaterally changing their strategy.  When applied to the soccer scenario, if the goalkeeper's best strategy is to guard the near post and, knowing this, the striker's best strategy is to shoot for the far post, in order for those strategies to form a Nash equilibrium neither actor should gain anything by changing his strategy arbitrarily.  The goal of Moschini's paper is to devise a game that allows one to deduce the Nash equilibrium for this soccer scenario and formulate an optimal strategy and expected goalscoring outcome. 

The model that Moschini devises is simple, but has a lot of notation.  Below right is a schematic of the goalkeeper vs. striker scenario.  The posts are marked 'F' and 'N', as you might expect, and the goalkeeper stands at a distance p from the far post.  So if p=0, he's at the far post, and if p=1, he's at the near post.  The striker shoots at one post or the other, and the probability that he will choose the near post is q, and the probability that he will choose the far post is 1-q.

Obviously the objective of the striker is to score a goal, and the objective of the 'keeper is to prevent one.  Moschini takes the following parameters into account for shot accuracy:

• distance
• skill
• field position (angle to the goal)

The following parameters are used to factored into the goalkeeper's success probability, which is inversely related to the striker's success probability:

• reaction ability
• position relative to posts

The result is a zero-sum sequential game — there's only one winner, and each player makes his choice after observing the other.  In most cases, the 'keeper guards a post, and the striker takes his shot.

I'm going to skip over the math where Moschini defines the payoffs to the striker for aiming at a particular post.  What's interesting is the results that he obtains when solving for Nash equilibria.  He shows that there is no equilibria when the striker takes a pure strategy; that is, he always aims at the near post or the far post.  (This result actually makes sense and is consistent with other games related to soccer, see the matching pennies problem.)  But when the striker tries to aim somewhere between the near and far posts, there does exist an equilibrium, which is called a mixed Nash equilibrium.  Moschini finds that the optimum strategy of the striker and 'keeper depends solely on the striker's location on the field, which in turn affects the accuracy of the shot.  I'd argue that the accuracy of the shot also depends on the striker's skill which isn't easily separable from distance, and that's true.  But in Moschini's simple formulation of the striker's scoring probability, the skill factor drops out. 

The optimum strategy for winning the 'game' is this:

• If you're the goalkeeper, guard the near post.
• If you're the striker, shoot for the far post.

And that strategy just happens to be consistent with the conventional wisdom of what goalkeepers and strikers should do.  We have an expectation; is that expectation consistent with what happens in a match?

To answer that question, Moschini tabulated goals scored during the 2002-03 Italian Serie A season.  He and his students watched the review shows from every round (what a job!), recorded every goal and the manner in which it was scored, and focused on goals that were scored toward the near or far post.  About two-thirds of these goals were scored at the far post, and one-third at the near post.  For this season, the data gave statistical support to the mixed Nash equilibria, but did not conclusively prove its existence. 

The academic in me likes this paper.  It's an enjoyable read and the reader won't suffer death by over-notation.  (I wish that were true on some of my papers!)  With the advances made in video analysis of soccer games, it would be easier to obtain more data to provide support for the mixed Nash equilibria.  However, as Moschini said in his findings, the current analysis doesn't allow one to evaluate changes in success probablility when the strategy changes — which would be sufficient to prove a Nash equilibrium. 

As much as I like the paper for academic reasons, I don't see how one could devise a metric that would be useful for evaluating striker performance.  It seems that those types of metrics would have near-post/far-post performance baked in.  Perhaps the analysis could be useful for managers when  evaluating player performance over a season; with the various video software one could compile a summary of near-post/far-post attacks on goal with corresponding success rates.  I guess I could see that being useful; it would be a tool for the technical staff to show the player what to work on during practice.  So in the end, it might be useful for internal study at the clubs, but for us desk soccermetricians developing such a metric would be too difficult.

As I said, comments are welcome.  I hope I got the game theory information right — I know enough to be dangerous, but I definitely wouldn't call myself an expert.  Here's a good introductory text on the subject.